A Concise Introduction to the Statistical Physics of Complex Systems

85 Pages·2012·0.527 MB·English

by  Bertin E.

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Eric Bertin
A Concise Introduction
to the Statistical Physics
of Complex Systems
123
Eric Bertin
Laboratoire de Physique
Ecole Normale Supérieurede Lyon
46allée d’Italie
69007Lyon
France
e-mail: [email protected]
ISSN 2191-5326 e-ISSN 2191-5334
ISBN 978-3-642-23922-9 e-ISBN978-3-642-23923-6
DOI 10.1007/978-3-642-23923-6
SpringerHeidelbergDordrechtLondonNewYork
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Preface
Inrecentyears,statisticalphysicsstartedraisingtheinterestofabroadcommunity
of researcher in the field of complex system sciences, ranging from biology to
social sciences, economics or computer sciences. More generally, a growing
numberofgraduatestudentsandresearchersfeeltheneedforlearningsomebasic
concepts and questions coming from other disciplines, leading for instance to the
organization of recurrent interdisciplinary summer schools.
The present booklet is partly based on the introductory lecture on statistical
physics given at the French Summer School on Complex Systems held both in
Lyon and Paris during the summers 2008 and 2009, and jointly organized by two
French Complex Systems Institutes, the ‘‘Institut des Systèmes Complexes Paris
IledeFrance’’(ISC-PIF)andthe‘‘InstitutRhône-AlpindesSystèmesComplexes’’
(IXXI). This introductory lecture was aimed at providing the participants with a
basic knowledge of the concepts and methods of statistical physics so that they
could later on follow more advanced lectures on diverse topics in the field of
complex systems. The lecture has been further extended in the framework of the
second year of Master in ‘‘Complex Systems Modelling’’ of the Ecole Normale
Supérieure de Lyon and Université Lyon 1, whose courses take place at IXXI.
It is a pleasure to thank Guillaume Beslon, Tommaso Roscilde and Sébastian
Grauwin,whowerealsoinvolvedinsomeofthelecturesmentionedabove,aswell
as Pablo Jensen for his efforts in setting up an interdisciplinary Master course on
complex systems, and for the fruitful collaboration we had over the last years.
v
Contents
1 Equilibrium Statistical Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Microscopic Dynamics of a Physical System . . . . . . . . . . . . . . 1
1.1.1 A Particle Attached to a Spring . . . . . . . . . . . . . . . . . . 1
1.1.2 Many-Particle System . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Case of Discrete Variables: Spin Models. . . . . . . . . . . . 5
1.2 Statistical Description of an Isolated System at Equilibrium. . . . 6
1.2.1 Notion of Statistical Description: A Toy Model . . . . . . . 6
1.2.2 Fundamental Postulate of Equilibrium
Statistical Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Computation of X(E) and S(E): Some
Simple Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Distribution of Energy Over Subsystems
and Statistical Temperature . . . . . . . . . . . . . . . . . . . . . 10
1.3 Equilibrium System in Contact with its Environment. . . . . . . . . 12
1.3.1 Exchanges of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Canonical Entropy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.3 Exchanges of Particles with a Reservoir:
The Grand-Canonical Ensemble . . . . . . . . . . . . . . . . . . 17
1.4 Phase Transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Example of the Ising Model. . . . . . . . . . . . . . . . . . . . . 18
1.4.2 Ising Model in Fully Connected Geometry. . . . . . . . . . . 18
1.4.3 Ising Model with Finite Connectivity . . . . . . . . . . . . . . 21
1.4.4 Renormalization Group Approach:
A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Disordered Systems and Glass Transition. . . . . . . . . . . . . . . . . 27
1.5.1 Disorder in Complex Systems: From Social Sciences
to Spin-Glasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
vii
viii Contents
1.5.2 Theoretical Spin-Glass Models. . . . . . . . . . . . . . . . . . . 28
1.5.3 The Simplest Disordered System: The Random
Energy Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Non-Stationary Dynamics and Stochastic Formalism. . . . . . . . . . . 33
2.1 Markovian Stochastic Processes and Master Equation . . . . . . . . 33
2.1.1 Definition of Markovian Stochastic Processes. . . . . . . . . 33
2.1.2 Master Equation and Detailed Balance . . . . . . . . . . . . . 35
2.1.3 Dynamical Increase of the Entropy. . . . . . . . . . . . . . . . 37
2.1.4 A Simple Example: The One-Dimensional
Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Langevin and Fokker–Planck Equations. . . . . . . . . . . . . . . . . . 41
2.2.1 Phenomenological Approach to the
Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2.2 Relation to Random Walks . . . . . . . . . . . . . . . . . . . . . 45
2.2.3 Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Anomalous Diffusion and Physical Aging . . . . . . . . . . . . . . . . 49
2.3.1 Generalized Central Limit Theorem . . . . . . . . . . . . . . . 49
2.3.2 Anomalous Diffusion. . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.3 Aging Dynamics and Trap Models . . . . . . . . . . . . . . . . 54
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Statistical Physics of Interacting Macroscopic ‘‘Entities’’. . . . . . . . 59
3.1 Dynamics of Residential Moves . . . . . . . . . . . . . . . . . . . . . . . 60
3.1.1 A Simplified Version of the Schelling Model. . . . . . . . . 61
3.1.2 Equilibrium Configurations of the Model. . . . . . . . . . . . 61
3.1.3 Condition for Phase Separation. . . . . . . . . . . . . . . . . . . 63
3.2 Condensation Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2.1 Zero Range Process: Definition and Exact Solution . . . . 66
3.2.2 Maximal Density and Condensation Phenomenon. . . . . . 66
3.3 Synchronization Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 The Kuramoto Model of Coupled Oscillators. . . . . . . . . 68
3.3.2 Synchronized Steady State. . . . . . . . . . . . . . . . . . . . . . 70
3.4 Collective Motion of Active Particles . . . . . . . . . . . . . . . . . . . 72
3.4.1 Definition of the Model. . . . . . . . . . . . . . . . . . . . . . . . 73
3.4.2 Description Through a Boltzmann Equation. . . . . . . . . . 73
3.4.3 Hydrodynamic Equations and Phase Diagram. . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Introduction
Generallyspeaking,thegoalsofstatisticalphysicsmaybesummarizedasfollows:
on the one hand to study systems composed of a large number of interacting
‘entities’,andontheotherhandtopredictthemacroscopic(orcollective)behavior
of the system considered from the microscopic laws ruling the dynamics of the
individual ‘entities’. These two goals are, to some extent, also shared by what is
nowadayscalled‘complexsystemsscience’.However,thespecificityofstatistical
physics is that:
• The ‘entities’ considered are in most cases atoms or molecules, for which the
individualmicroscopiclawsareknownfromfundamentalphysicaltheories—at
variancewithotherfieldslikesocialsciencesforexample,wherelittleisknown
about the quantitative behavior of individuals.
• These atoms, or molecules, are often all of the same type, or at most of a few
different types—in contrast to biological or social systems for instance, where
the individual ‘entities’ may all differ, or at least belong to a large number of
different types.
For these reasons, systems studied in the framework of statistical physics may
be considered as among the simplest examples of complex systems. One further
specificityofstatisticalphysicswithrespecttoothersciencesaimingatdescribing
the collective behavior of complex systems is that it allows for a rather well-
developed mathematical treatment.
The present booklet is divided into three chapters. The first one deals with
equilibriumstatisticalphysics,tryingtoexposeinaconcisewaythemainconcepts
of this theory, and paying specific attention to those concepts that could be more
generally relevant to complex system sciences. The second part mainly aims at
describingtime-dependenteffects(occurringforinstancewhenasystemrelaxesto
equilibrium from some non-equilibrium initial condition) in the framework of
stochastic markovian processes. Emphasis is put on simple and generic models,
and some relations to probability theory are also outlined. Finally, the third part
presents a few examples of applications of statistical physics to other types of
ix
x Introduction
complex systems, beyond the strict realm of physics, with the hope to trigger the
interest of readers coming from various disciplinary fields. Simple models of
systems composed of a large number of macroscopic ‘entities’ that do not follow
thesamelawsasphysicalatomsormolecules(forinstancesandgrains,animalsor
social agents) are considered. As no general statistical framework exists for such
systems, their description relies on the case-by-case adaptation of different tech-
niques borrowed from standard statistical physics, ranging from mappings to
effective equilibrium systems, to Boltzmann approaches (a technique early
developedinstatisticalphysicstocharacterizethedynamicsofgases)forsystems
interacting through binary collisions, or to exact solutions when available.